This paper develops a Lagrangian dual-based branch-and-bound algorithm for the generalized multi-assignment problem (GMAP) which includes the well-known generalized assignment problem (GAP) as a special case. In GMAP, an object may be required to be duplicated in multiple locations. We develop a Lagrangian dual ascent algorithm for GMAP. This dual ascent and the subgradient search each possess advantages that can be combined to develop a new Lagrangian dual search algorithm. The latter algorithm, when incorporated into a branch-and-bound algorithm as the lower bounding scheme, can accelerate the search process. Computational results demonstrate the efficiency and robustness of this branch-and-bound algorithm not only for GMAPs, but for GAPs that are more difficult than could be solved by previous algorithms.Generalized Multi-Assignment Problem, Generalized Assignment Problem, Lagrangian Dual Ascent, Subgradient Search, Lagrangian Dual-Based Branch and Bound
We address the physical SONET network design problem of selecting stackable, unidirectional rings connecting central office nodes (COs) and remote nodes (RNs). This problem frequently arises in designing feeder transport networks to support centralized traffic between the COs and RNs. We formulate a 0-1 programming model for this problem. A simulated annealing-based Lagrangian relaxation procedure to find optimal or near-optimal solutions is then described. Computational results are reported showing that our procedures produce solutions that are on average within 1.1% of optimality. We show that using simulated annealing to augment the pure Lagrangian approach produces superior solutions to the Lagrangian approach.
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